{"history":[],"article":{"id":"96e4c513-24fb-4e96-85f3-a0d304cf1739","title":"Polar coordinate system","slug":"polar-coordinate-system","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **polar [coordinate system](https://bliptext.com/articles/coordinate-system)** is a [two-dimensional](https://bliptext.com/articles/two-dimensional) [coordinate system](https://bliptext.com/articles/coordinate-system) in which each [point](https://bliptext.com/articles/point-mathematics) on a [plane](https://bliptext.com/articles/plane-mathematics) is determined by a [distance](https://bliptext.com/articles/distance) from a reference [point](https://bliptext.com/articles/point-mathematics) and an [angle](https://bliptext.com/articles/angle) from a reference direction.\nThe reference point (analogous to the origin of a [Cartesian coordinate system](https://bliptext.com/articles/cartesian-coordinate-system)) is called the pole, and the [ray](https://bliptext.com/articles/ray-geometry) from the pole in the reference direction is the polar axis.\nThe distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or [azimuth](https://bliptext.com/articles/azimuth).\nAngles in polar notation are generally expressed in either [degrees](https://bliptext.com/articles/degree-angle) or [radians](https://bliptext.com/articles/radian) ([π](https://bliptext.com/articles/pi) rad being equal to 180° and 2[π](https://bliptext.com/articles/pi) rad being equal to 360°).\n[Grégoire de Saint-Vincent](https://bliptext.com/articles/gr-goire-de-saint-vincent) and [Bonaventura Cavalieri](https://bliptext.com/articles/bonaventura-cavalieri) independently introduced the concepts in the mid-17th century, though the actual term \"polar coordinates\" has been attributed to [Gregorio Fontana](https://bliptext.com/articles/gregorio-fontana) in the 18th century.\nThe initial motivation for the introduction of the polar system was the study of [circular](https://bliptext.com/articles/circular-motion) and [orbital motion](https://bliptext.com/articles/orbital-motion).\nPolar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as [spirals](https://bliptext.com/articles/spiral).\nPlanar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.\nThe polar coordinate system is extended to three dimensions in two ways: the [cylindrical](https://bliptext.com/articles/cylindrical-coordinate-system) and [spherical](https://bliptext.com/articles/spherical-coordinate-system) coordinate systems.\n# [Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°). | upload.wikimedia.org/wikipedia/commons/d/d3/Examples///of///Polar///Coordinates.svg]\n\n# History\nThe concepts of angle and radius were already used by ancient peoples of the first millennium [BC](https://bliptext.com/articles/before-christ).\nThe [Greek astronomer](https://bliptext.com/articles/greek-astronomy) and [astrologer](https://bliptext.com/articles/astrologer) [Hipparchus](https://bliptext.com/articles/hipparchus) (190–120 BC) created a table of [chord](https://bliptext.com/articles/chord-geometry) functions giving the length of the [chord](https://bliptext.com/articles/chord-geometry) for each angle, and there are references to his using polar coordinates in establishing stellar positions.\nIn [On Spirals](https://bliptext.com/articles/on-spirals), [Archimedes](https://bliptext.com/articles/archimedes) describes the [Archimedean spiral](https://bliptext.com/articles/archimedean-spiral), a function whose radius depends on the angle.\nThe Greek work, however, did not extend to a full coordinate system.\nFrom the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to [Mecca](https://bliptext.com/articles/mecca) ([qibla](https://bliptext.com/articles/qibla))—and its distance—from any location on the Earth.\nFrom the 9th century onward they were using [spherical trigonometry](https://bliptext.com/articles/spherical-trigonometry) and [map projection](https://bliptext.com/articles/map-projection) methods to determine these quantities accurately.\nThe calculation is essentially the conversion of the [equatorial polar coordinates](https://bliptext.com/articles/geodetic-coordinates) of Mecca (i.e. its [longitude](https://bliptext.com/articles/longitude) and [latitude](https://bliptext.com/articles/latitude)) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the [great circle](https://bliptext.com/articles/great-circle) through the given location and the Earth's poles and whose polar axis is the line through the location and its [antipodal point](https://bliptext.com/articles/antipodal-point).\nThere are various accounts of the introduction of polar coordinates as part of a formal coordinate system.\nThe full history of the subject is described in [Harvard](https://bliptext.com/articles/harvard-university) professor [Julian Lowell Coolidge](https://bliptext.com/articles/julian-lowell-coolidge)'s Origin of Polar Coordinates. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century.\nSaint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653.\nCavalieri first used polar coordinates to solve a problem relating to the area within an [Archimedean spiral](https://bliptext.com/articles/archimedean-spiral).\n[Blaise Pascal](https://bliptext.com/articles/blaise-pascal) subsequently used polar coordinates to calculate the length of [parabolic arcs](https://bliptext.com/articles/parabola).\nIn [Method of Fluxions](https://bliptext.com/articles/method-of-fluxions) (written 1671, published 1736), Sir [Isaac Newton](https://bliptext.com/articles/isaac-newton) examined the transformations between polar coordinates, which he referred to as the \"Seventh Manner; For Spirals\", and nine other coordinate systems.\nIn the journal [Acta Eruditorum](https://bliptext.com/articles/acta-eruditorum) (1691), [Jacob Bernoulli](https://bliptext.com/articles/jacob-bernoulli) used a system with a point on a line, called the pole and polar axis respectively.\nCoordinates were specified by the distance from the pole and the angle from the polar axis.\nBernoulli's work extended to finding the [radius of curvature](https://bliptext.com/articles/radius-of-curvature-mathematics) of curves expressed in these coordinates.\nThe actual term polar coordinates has been attributed to [Gregorio Fontana](https://bliptext.com/articles/gregorio-fontana) and was used by 18th-century Italian writers.\nThe term appeared in [English](https://bliptext.com/articles/english-language) in [George Peacock](https://bliptext.com/articles/george-peacock-mathematician)'s 1816 translation of [Lacroix](https://bliptext.com/articles/sylvestre-fran-ois-lacroix)'s Differential and Integral Calculus.\n[Alexis Clairaut](https://bliptext.com/articles/alexis-clairaut) was the first to think of polar coordinates in three dimensions, and [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) was the first to actually develop them.\n# [Hipparchus | upload.wikimedia.org/wikipedia/commons/c/c6/Head///of///Hipparchus///%28cropped%29.jpg]\n\n# Conventions\nThe radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t.\nThe angular coordinate is specified as φ by [ISO](https://bliptext.com/articles/international-organization-for-standardization) standard [31-11](https://bliptext.com/articles/iso-31-11).\nHowever, in mathematical literature the angle is often denoted by θ instead.\nAngles in polar notation are generally expressed in either [degrees](https://bliptext.com/articles/degree-angle) or [radians](https://bliptext.com/articles/radian) (2[π](https://bliptext.com/articles/pi) rad being equal to 360°).\nDegrees are traditionally used in [navigation](https://bliptext.com/articles/navigation), [surveying](https://bliptext.com/articles/surveying), and many applied disciplines, while radians are more common in mathematics and mathematical [physics](https://bliptext.com/articles/physics).\nThe angle φ is defined to start at 0° from a reference direction, and to increase for rotations in either [clockwise (cw)](https://bliptext.com/articles/clockwise) or counterclockwise (ccw) orientation.\nFor example, in mathematics, the reference direction is usually drawn as a [ray](https://bliptext.com/articles/ray-geometry) from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation ([bearing](https://bliptext.com/articles/bearing-navigation), [heading](https://bliptext.com/articles/heading-navigation)) the 0°-[heading](https://bliptext.com/articles/heading-navigation) is drawn vertically upwards and the angle increases for cw rotations.\nThe polar angles decrease towards negative values for rotations in the respectively opposite orientations.\n# [A polar grid with several angles, increasing in counterclockwise orientation and labelled in degrees | upload.wikimedia.org/wikipedia/commons/1/13/Polar///graph///paper.svg]\n\n## Uniqueness of polar coordinates\nAdding any number of full [turns](https://bliptext.com/articles/turn-geometry) (360°) to the angular coordinate does not change the corresponding direction.\nSimilarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle).\nTherefore, the same point (r, φ) can be expressed with an infinite number of different polar coordinates (r, φ + n × 360°) and (−r, φ + 180° + n × 360°) = (−r, φ + (2n + 1) × 180°), where n is an arbitrary [integer](https://bliptext.com/articles/integer).\nMoreover, the pole itself can be expressed as (0, φ) for any angle φ.\nWhere a unique representation is needed for any point besides the pole, it is usual to limit r to positive numbers (r > 0) and φ to either the [interval](https://bliptext.com/articles/interval-mathematics) or the [interval](https://bliptext.com/articles/interval-mathematics), which in radians are or.\nAnother convention, in reference to the usual [codomain](https://bliptext.com/articles/codomain) of the [arctan function](https://bliptext.com/articles/inverse-trigonometric-functions), is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to.\nIn all cases a unique azimuth for the pole (r = 0) must be chosen, e.g., φ = 0.\n\n# Converting between polar and Cartesian coordinates\nThe polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) sine and cosine:\nThe Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by:\nwhere hypot is the [Pythagorean sum](https://bliptext.com/articles/pythagorean-addition) and [atan2](https://bliptext.com/articles/atan2) is a common variation on the [arctangent](https://bliptext.com/articles/arctangent) function defined as\nIf r is calculated first as above, then this formula for φ may be stated more simply using the [arccosine](https://bliptext.com/articles/arccosine) function:\n# [A diagram illustrating the relationship between polar and Cartesian coordinates. | upload.wikimedia.org/wikipedia/commons/7/78/Polar///to///cartesian.svg]\n\n## Complex numbers\nEvery [complex number](https://bliptext.com/articles/complex-number) can be represented as a point in the [complex plane](https://bliptext.com/articles/complex-plane), and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form).\nIn polar form, the distance and angle coordinates are often referred to as the number's **magnitude** and **argument** respectively.\nTwo complex numbers can be multiplied by adding their arguments and multiplying their magnitudes.\nThe complex number z can be represented in rectangular form as z = x + iy where i is the [imaginary unit](https://bliptext.com/articles/imaginary-unit), or can alternatively be written in polar form as\nand from there, by [Euler's formula](https://bliptext.com/articles/euler-s-formula), as\nwhere e is [Euler's number](https://bliptext.com/articles/e-mathematical-constant), and φ, expressed in radians, is the [principal value](https://bliptext.com/articles/principal-value) of the complex number function [arg](https://bliptext.com/articles/argument-complex-analysis) applied to x + iy.\nTo convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used.\nEquivalent are the\ncis\nand [angle notations](https://bliptext.com/articles/angle-notation):\nFor the operations of [multiplication](https://bliptext.com/articles/multiplication), [division](https://bliptext.com/articles/division-mathematics), [exponentiation](https://bliptext.com/articles/exponentiation), and [root extraction](https://bliptext.com/articles/root-extraction) of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form.\nFrom the laws of exponentiation:\n# [An illustration of a complex number z plotted on the complex plane | upload.wikimedia.org/wikipedia/commons/7/71/Imaginarynumber2.svg]\n# [An illustration of a complex number plotted on the complex plane using Euler's formula | upload.wikimedia.org/wikipedia/commons/7/71/Euler%27s///formula.svg]\n\n# Polar equation of a curve\nThe equation defining a [plane curve](https://bliptext.com/articles/plane-curve) expressed in polar coordinates is known as a polar equation.\nIn many cases, such an equation can simply be specified by defining r as a [function](https://bliptext.com/articles/function-mathematics) of φ.\nThe resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the [graph](https://bliptext.com/articles/graph-of-a-function) of the polar function r.\nNote that, in contrast to Cartesian coordinates, the independent variable φ is the second entry in the ordered pair.\nDifferent forms of [symmetry](https://bliptext.com/articles/symmetry) can be deduced from the equation of a polar function r:\nr(−φ) = r(φ)\nthe curve will be symmetrical about the horizontal (0°/180°) ray;\nr(π − φ) = r(φ)\nit will be symmetric about the vertical (90°/270°) ray:\nr(φ − α) = r(φ)\nit will be [rotationally symmetric](https://bliptext.com/articles/rotational-symmetry) by α clockwise and counterclockwise about the pole.\nBecause of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate.\nAmong the best known of these curves are the [polar rose](https://bliptext.com/articles/rose-mathematics), [Archimedean spiral](https://bliptext.com/articles/archimedean-spiral), [lemniscate](https://bliptext.com/articles/lemniscate-of-bernoulli), [limaçon](https://bliptext.com/articles/lima-on), and [cardioid](https://bliptext.com/articles/cardioid).\nFor the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.\n\n## Circle\n[[Image:circle r=1.svg|thumb|right|A circle with equation\nr(φ) = 1\n]] The general equation for a circle with a center at and radius a is\nThis can be simplified in various ways, to conform to more specific cases, such as the equation\nfor a circle with a center at the pole and radius a.\nWhen\nr0 = a\nor the origin lies on the circle, the equation becomes\nIn the general case, the equation can be solved for\nr\n, giving\nThe solution with a minus sign in front of the square root gives the same curve.\n\n## Line\nRadial lines (those running through the pole) are represented by the equation\nwhere \\gamma is the angle of elevation of the line; that is,, where m is the [slope](https://bliptext.com/articles/slope) of the line in the Cartesian coordinate system.\nThe non-radial line that crosses the radial line [perpendicularly](https://bliptext.com/articles/perpendicular) at the point has the equation\nOtherwise stated is the point in which the tangent intersects the imaginary circle of radius r_0\n\n## Polar rose\n[[Image:Rose 2sin(4theta).svg|thumb|right|A polar rose with equation\nr(φ) = 2 sin 4φ\n]] A [polar rose](https://bliptext.com/articles/rose-mathematics) is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,\nfor any constant γ0 (including 0).\nIf k is an integer, these equations will produce a k-petaled rose if k is [odd](https://bliptext.com/articles/even-and-odd-numbers), or a 2k-petaled rose if k is even.\nIf k is rational, but not an integer, a rose-like shape may form but with overlapping petals.\nNote that these equations never define a rose with 2, 6, 10, 14, etc. petals.\nThe [variable](https://bliptext.com/articles/variable-math) a directly represents the length or amplitude of the petals of the rose, while k relates to their spatial frequency.\nThe constant γ0 can be regarded as a phase angle.\n\n## Archimedean spiral\n[[File:Spiral of Archimedes.svg|thumb|right|One arm of an Archimedean spiral with equation\nr(φ) = φ / 2π\nfor\n0 < φ < 6π\n]] The [Archimedean spiral](https://bliptext.com/articles/archimedean-spiral) is a spiral discovered by [Archimedes](https://bliptext.com/articles/archimedes) which can also be expressed as a simple polar equation.\nIt is represented by the equation\nChanging the parameter a will turn the spiral, while b controls the distance between the arms, which for a given spiral is always constant.\nThe Archimedean spiral has two arms, one for\nφ > 0\nand one for\nφ < 0\n. The two arms are smoothly connected at the pole.\nIf\na = 0\n, taking the mirror image of one arm across the 90°/270° line will yield the other arm.\nThis curve is notable as one of the first curves, after the [conic sections](https://bliptext.com/articles/conic-section), to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation.\n# [Ellipse, showing semi-latus rectum | upload.wikimedia.org/wikipedia/commons/3/35/Elps-slr.svg]\n\n## Conic sections\nA [conic section](https://bliptext.com/articles/conic-section) with one focus on the pole and the other somewhere on the 0° ray (so that the conic's [major axis](https://bliptext.com/articles/semi-major-axis) lies along the polar axis) is given by:\nwhere e is the [eccentricity](https://bliptext.com/articles/eccentricity-mathematics) and \\ell is the [semi-latus rectum](https://bliptext.com/articles/semi-latus-rectum) (the perpendicular distance at a focus from the major axis to the curve).\nIf e > 1, this equation defines a [hyperbola](https://bliptext.com/articles/hyperbola); if\ne = 1\n, it defines a [parabola](https://bliptext.com/articles/parabola); and if\ne < 1\n, it defines an [ellipse](https://bliptext.com/articles/ellipse).\nThe special case\ne = 0\nof the latter results in a circle of the radius \\ell.\n\n## Quadratrix\nA quadratrix in the first quadrant (x, y) is a curve with y = ρ sin θ equal to the fraction of the quarter circle with radius r determined by the radius through the curve point.\nSince this fraction is, the curve is given by.\n\n## Intersection of two polar curves\nThe graphs of two polar functions and have possible intersections of three types:\n\n# Calculus\n[Calculus](https://bliptext.com/articles/calculus) can be applied to equations expressed in polar coordinates.\nThe angular coordinate φ is expressed in radians throughout this section, which is the conventional choice when doing calculus.\n\n## Differential calculus\nUsing\nx = r cos φ\nand\ny = r sin φ\n, one can derive a relationship between derivatives in Cartesian and polar coordinates.\nFor a given function, u(x,y), it follows that (by computing its [total derivatives](https://bliptext.com/articles/total-derivative))\nor\nHence, we have the following formula:\nUsing the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives.\nGiven a function u(r,φ), it follows that\nor\nHence, we have the following formulae:\nTo find the Cartesian slope of the tangent line to a polar curve r(φ) at any given point, the curve is first expressed as a system of [parametric equations](https://bliptext.com/articles/parametric-equations).\n[Differentiating](https://bliptext.com/articles/derivative) both equations with respect to φ yields\nDividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(φ), φ):\nFor other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see [curvilinear coordinates](https://bliptext.com/articles/curvilinear-coordinates).\n\n## Integral calculus (arc length)\nThe arc length (length of a line segment) defined by a polar function is found by the integration over the curve r(φ).\nLet L denote this length along the curve starting from points A through to point B, where these points correspond to φ = a and φ = b such that\n0 < b − a < 2π\n. The length of L is given by the following integral\n\n## Integral calculus (area)\nLet R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π.\nThen, the area of R is\nThis result can be found as follows.\nFirst, the interval [a, b] is divided into n subintervals, where n is some positive integer.\nThus Δφ, the angle measure of each subinterval, is equal to\nb − a\n(the total angle measure of the interval), divided by n, the number of subintervals.\nFor each subinterval i = 1, 2, ..., n, let φi be the midpoint of the subinterval, and construct a [sector](https://bliptext.com/articles/circular-sector) with the center at the pole, radius r(φi), central angle Δφ and arc length r(φi)Δφ.\nThe area of each constructed sector is therefore equal to\nHence, the total area of all of the sectors is\nAs the number of subintervals n is increased, the approximation of the area improves.\nTaking n → ∞, the sum becomes the [Riemann sum](https://bliptext.com/articles/riemann-sum) for the above integral.\nA mechanical device that computes area integrals is the [planimeter](https://bliptext.com/articles/planimeter), which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element [linkage](https://bliptext.com/articles/linkage-mechanical) effects [Green's theorem](https://bliptext.com/articles/green-s-theorem), converting the quadratic polar integral to a linear integral.\n# [polar coordinate system | upload.wikimedia.org/wikipedia/commons/4/4c/Polar///coordinates///integration///region.svg]\n# [The region R is approximated by n sectors (here, n = 5). | upload.wikimedia.org/wikipedia/commons/4/4c/Polar///coordinates///integration///Riemann///sum.svg]\n# [A planimeter, which mechanically computes polar integrals | upload.wikimedia.org/wikipedia/commons/5/54/Planimeter.jpg]\n\n### Generalization\nUsing [Cartesian coordinates](https://bliptext.com/articles/cartesian-coordinates), an infinitesimal area element can be calculated as dA = dx dy.\nThe [substitution rule for multiple integrals](https://bliptext.com/articles/integration-by-substitution) states that, when using other coordinates, the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant) determinant of the coordinate conversion formula has to be considered:\nHence, an area element in polar coordinates can be written as\nNow, a function, that is given in polar coordinates, can be integrated as follows:\nHere, R is the same region as above, namely, the region enclosed by a curve r(φ) and the rays φ = a and φ = b.\nThe formula for the area of R is retrieved by taking f identically equal to 1.\nA more surprising application of this result yields the [Gaussian integral](https://bliptext.com/articles/gaussian-integral):\n# [A graph of and the area between the function and the x-axis, which is equal to \\sqrt{\\pi}. | upload.wikimedia.org/wikipedia/commons/2/2f/E%5E%28-x%5E2%29.svg]\n\n## Vector calculus\n[Vector calculus](https://bliptext.com/articles/vector-calculus) can also be applied to polar coordinates.\nFor a planar motion, let \\mathbf{r} be the position vector\n(r cos(φ), r sin(φ))\n, with r and\nφ\ndepending on time t.\nWe define an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) with three unit vectors: radial, transverse, and normal directions.\nThe radial direction is defined by normalizing \\mathbf{r}:\nRadial and velocity directions span the plane of the motion, whose normal direction is denoted :\nThe transverse direction is perpendicular to both radial and normal directions:\nThen\nThis equation can be obtain by taking derivative of the function and derivatives of the unit basis vectors.\nFor a curve in 2D where the parameter is \\theta the previous equations simplify to:\n\n### Centrifugal and Coriolis terms\nThe term is sometimes referred to as the centripetal acceleration, and the term as the Coriolis acceleration.\nFor example, see Shankar.\nNote: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used.\nIn planar particle dynamics these accelerations appear when setting up Newton's [second law of motion](https://bliptext.com/articles/newton-s-second-law) in a rotating frame of reference.\nHere these extra terms are often called [fictitious forces](https://bliptext.com/articles/fictitious-force); fictitious because they are simply a result of a change in coordinate frame.\nThat does not mean they do not exist, rather they exist only in the rotating frame.\n# [Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′ at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ϕ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path. | upload.wikimedia.org/wikipedia/commons/0/0c/Co-rotating///frame///vector.svg]\n\n#### Co-rotating frame\nFor a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous co-rotating frame of reference.\nTo define a co-rotating frame, first an origin is selected from which the distance r(t) to the particle is defined.\nAn axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin.\nThen, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dφ/dt.\nNext, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame.\nLet the location of the particle in the inertial frame be (r(t), φ(t)), and in the co-rotating frame be (r′(t), φ′(t)'').\nBecause the co-rotating frame rotates at the same rate as the particle, dφ′/dt = 0.\nThe fictitious centrifugal force in the co-rotating frame is mrΩ2, radially outward.\nThe velocity of the particle in the co-rotating frame also is radially outward, because dφ′/dt = 0.\nThe fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing φ only.\nThus, using these forces in Newton's second law we find:\nwhere over dots represent derivatives with respect to time, and **F** is the net real force (as opposed to the fictitious forces).\nIn terms of components, this vector equation becomes:\nwhich can be compared to the equations for the inertial frame:\nThis comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dφ/dt, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.\nFor general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous [osculating circle](https://bliptext.com/articles/osculating-circle) of its motion, not to a fixed center of polar coordinates.\nFor more detail, see [centripetal force](https://bliptext.com/articles/centripetal-force).\n\n# Differential geometry\nIn the modern terminology of [differential geometry](https://bliptext.com/articles/differential-geometry), polar coordinates provide [coordinate charts](https://bliptext.com/articles/coordinate-charts) for the [differentiable manifold](https://bliptext.com/articles/differentiable-manifold)\nR2 \\ {(0,0)}\n, the plane minus the origin.\nIn these coordinates, the Euclidean [metric tensor](https://bliptext.com/articles/metric-tensor) is given byThis can be seen via the change of variables formula for the [metric tensor](https://bliptext.com/articles/metric-tensor), or by computing the [differential forms](https://bliptext.com/articles/differential-form) dx, dy via the [exterior derivative](https://bliptext.com/articles/exterior-derivative) of the 0-forms\nx = r cos(θ)\n,\ny = r sin(θ)\nand substituting them in the Euclidean metric tensor\nds2 = dx2 + dy2\n.\nLet, and be two points in the plane given by their cartesian and polar coordinates.\nThen\nSince, and , we get that\nNow we use the trigonometric identity to proceed:\nIf the radial and angular quantities are near to each other, and therefore near to a common quantity r and \\theta, we have that.\nMoreover, the cosine of d\\theta can be approximated with the Taylor series of the cosine up to linear terms:\nso that, and.\nTherefore, around an infinitesimally small domain of any point,\nas stated.\nAn [orthonormal](https://bliptext.com/articles/orthonormality) [frame](https://bliptext.com/articles/moving-frame) with respect to this metric is given bywith dual co[frame](https://bliptext.com/articles/moving-frame)The [connection form](https://bliptext.com/articles/connection-form) relative to this [frame](https://bliptext.com/articles/moving-frame) and the [Levi-Civita connection](https://bliptext.com/articles/levi-civita-connection) is given by the skew-symmetric matrix of 1-formsand hence the [curvature form](https://bliptext.com/articles/curvature-form)\nΩ = dω + ω∧ω\nvanishes.\nTherefore, as expected, the punctured plane is a [flat manifold](https://bliptext.com/articles/flat-manifold).\n\n# Extensions in 3D\nThe polar coordinate system is extended into three dimensions with two different coordinate systems, the [cylindrical](https://bliptext.com/articles/cylindrical-coordinate-system) and [spherical coordinate system](https://bliptext.com/articles/spherical-coordinate-system).\n\n# Applications\nPolar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane.\nThey are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point.\nFor instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate.\nMoreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates.\nThe initial motivation for the introduction of the polar system was the study of [circular](https://bliptext.com/articles/circular-motion) and [orbital motion](https://bliptext.com/articles/orbital-motion).\n\n## Position and navigation\nPolar coordinates are used often in [navigation](https://bliptext.com/articles/navigation) as the destination or direction of travel can be given as an angle and distance from the object being considered.\nFor instance, [aircraft](https://bliptext.com/articles/aircraft) use a slightly modified version of the polar coordinates for navigation.\nIn this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system.\nHeading 360 corresponds to [magnetic north](https://bliptext.com/articles/magnetic-north), while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.\nThus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read [zero-niner-zero](https://bliptext.com/articles/icao-spelling-alphabet) by [air traffic control](https://bliptext.com/articles/air-traffic-control)).\n\n## Modeling\nSystems displaying [radial symmetry](https://bliptext.com/articles/radial-symmetry) provide natural settings for the polar coordinate system, with the central point acting as the pole.\nA prime example of this usage is the [groundwater flow equation](https://bliptext.com/articles/groundwater-flow-equation) when applied to radially symmetric wells.\nSystems with a [radial force](https://bliptext.com/articles/central-force) are also good candidates for the use of the polar coordinate system.\nThese systems include [gravitational fields](https://bliptext.com/articles/gravitation), which obey the [inverse-square law](https://bliptext.com/articles/inverse-square-law), as well as systems with [point sources](https://bliptext.com/articles/point-source), such as [radio antennas](https://bliptext.com/articles/antenna-radio).\nRadially asymmetric systems may also be modeled with polar coordinates.\nFor example, a [microphone](https://bliptext.com/articles/microphone)'s [pickup pattern](https://bliptext.com/articles/microphone-pick-up-patterns) illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves.\nThe curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as at its target design frequency.\nThe pattern shifts toward omnidirectionality at lower frequencies.\n\n## General references","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:23:00.617Z","updated_at":"2025-01-25T17:23:00.617Z","search_vector":"'coordin':2 'polar':1 'system':3","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **polar [coordinate system](https://bliptext.com/articles/coordinate-system)** is a [two-dimensional](https://bliptext.com/articles/two-dimensional) [coordinate system](https://bliptext.com/articles/coordinate-system) in which each [point](https://bliptext.com/articles/point-mathematics) on a [plane](https://bliptext.com/articles/plane-mathematics) is determined by a [distance](https://bliptext.com/articles/distance) from a reference [point](https://bliptext.com/articles/point-mathematics) and an [angle](https://bliptext.com/articles/angle) from a reference direction.\nThe reference point (analogous to the origin of a [Cartesian coordinate system](https://bliptext.com/articles/cartesian-coordinate-system)) is called the pole, and the [ray](https://bliptext.com/articles/ray-geometry) from the pole in the reference direction is the polar axis.\nThe distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or [azimuth](https://bliptext.com/articles/azimuth).\nAngles in polar notation are generally expressed in either [degrees](https://bliptext.com/articles/degree-angle) or [radians](https://bliptext.com/articles/radian) ([[π](https://bliptext.com/articles/pi)](https://bliptext.com/articles/pi) rad being equal to 180° and 2[[π](https://bliptext.com/articles/pi)](https://bliptext.com/articles/pi) rad being equal to 360°).\n[Grégoire de Saint-Vincent](https://bliptext.com/articles/gr-goire-de-saint-vincent) and [Bonaventura Cavalieri](https://bliptext.com/articles/bonaventura-cavalieri) independently introduced the concepts in the mid-17th century, though the actual term \"polar coordinates\" has been attributed to [Gregorio Fontana](https://bliptext.com/articles/gregorio-fontana) in the 18th century.\nThe initial motivation for the introduction of the polar system was the study of [circular](https://bliptext.com/articles/circular-motion) and [orbital motion](https://bliptext.com/articles/orbital-motion).\nPolar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as [spirals](https://bliptext.com/articles/spiral).\nPlanar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.\nThe polar coordinate system is extended to three dimensions in two ways: the [cylindrical](https://bliptext.com/articles/cylindrical-coordinate-system) and [spherical](https://bliptext.com/articles/spherical-coordinate-system) coordinate systems.\n# [Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°). | upload.wikimedia.org/wikipedia/commons/d/d3/Examples///of///Polar///Coordinates.svg]\n\n# History\nThe concepts of angle and radius were already used by ancient peoples of the first millennium [BC](https://bliptext.com/articles/before-christ).\nThe [Greek astronomer](https://bliptext.com/articles/greek-astronomy) and [astrologer](https://bliptext.com/articles/astrologer) [Hipparchus](https://bliptext.com/articles/hipparchus) (190–120 BC) created a table of [chord](https://bliptext.com/articles/chord-geometry) functions giving the length of the [chord](https://bliptext.com/articles/chord-geometry) for each angle, and there are references to his using polar coordinates in establishing stellar positions.\nIn [On Spirals](https://bliptext.com/articles/on-spirals), [Archimedes](https://bliptext.com/articles/archimedes) describes the [Archimedean spiral](https://bliptext.com/articles/archimedean-spiral), a function whose radius depends on the angle.\nThe Greek work, however, did not extend to a full coordinate system.\nFrom the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to [Mecca](https://bliptext.com/articles/mecca) ([qibla](https://bliptext.com/articles/qibla))—and its distance—from any location on the Earth.\nFrom the 9th century onward they were using [spherical trigonometry](https://bliptext.com/articles/spherical-trigonometry) and [map projection](https://bliptext.com/articles/map-projection) methods to determine these quantities accurately.\nThe calculation is essentially the conversion of the [equatorial polar coordinates](https://bliptext.com/articles/geodetic-coordinates) of Mecca (i.e. its [longitude](https://bliptext.com/articles/longitude) and [latitude](https://bliptext.com/articles/latitude)) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the [great circle](https://bliptext.com/articles/great-circle) through the given location and the Earth's poles and whose polar axis is the line through the location and its [antipodal point](https://bliptext.com/articles/antipodal-point).\nThere are various accounts of the introduction of polar coordinates as part of a formal coordinate system.\nThe full history of the subject is described in [Harvard](https://bliptext.com/articles/harvard-university) professor [Julian Lowell Coolidge](https://bliptext.com/articles/julian-lowell-coolidge)'s Origin of Polar Coordinates. Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century.\nSaint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653.\nCavalieri first used polar coordinates to solve a problem relating to the area within an [Archimedean spiral](https://bliptext.com/articles/archimedean-spiral).\n[Blaise Pascal](https://bliptext.com/articles/blaise-pascal) subsequently used polar coordinates to calculate the length of [parabolic arcs](https://bliptext.com/articles/parabola).\nIn [Method of Fluxions](https://bliptext.com/articles/method-of-fluxions) (written 1671, published 1736), Sir [Isaac Newton](https://bliptext.com/articles/isaac-newton) examined the transformations between polar coordinates, which he referred to as the \"Seventh Manner; For Spirals\", and nine other coordinate systems.\nIn the journal [Acta Eruditorum](https://bliptext.com/articles/acta-eruditorum) (1691), [Jacob Bernoulli](https://bliptext.com/articles/jacob-bernoulli) used a system with a point on a line, called the pole and polar axis respectively.\nCoordinates were specified by the distance from the pole and the angle from the polar axis.\nBernoulli's work extended to finding the [radius of curvature](https://bliptext.com/articles/radius-of-curvature-mathematics) of curves expressed in these coordinates.\nThe actual term polar coordinates has been attributed to [Gregorio Fontana](https://bliptext.com/articles/gregorio-fontana) and was used by 18th-century Italian writers.\nThe term appeared in [English](https://bliptext.com/articles/english-language) in [George Peacock](https://bliptext.com/articles/george-peacock-mathematician)'s 1816 translation of [Lacroix](https://bliptext.com/articles/sylvestre-fran-ois-lacroix)'s Differential and Integral Calculus.\n[Alexis Clairaut](https://bliptext.com/articles/alexis-clairaut) was the first to think of polar coordinates in three dimensions, and [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) was the first to actually develop them.\n# [Hipparchus | upload.wikimedia.org/wikipedia/commons/c/c6/Head///of///Hipparchus///%28cropped%29.jpg]\n\n# Conventions\nThe radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t.\nThe angular coordinate is specified as φ by [ISO](https://bliptext.com/articles/international-organization-for-standardization) standard [31-11](https://bliptext.com/articles/iso-31-11).\nHowever, in mathematical literature the angle is often denoted by θ instead.\nAngles in polar notation are generally expressed in either [degrees](https://bliptext.com/articles/degree-angle) or [radians](https://bliptext.com/articles/radian) (2[π](https://bliptext.com/articles/pi) rad being equal to 360°).\nDegrees are traditionally used in [navigation](https://bliptext.com/articles/navigation), [surveying](https://bliptext.com/articles/surveying), and many applied disciplines, while radians are more common in mathematics and mathematical [physics](https://bliptext.com/articles/physics).\nThe angle φ is defined to start at 0° from a reference direction, and to increase for rotations in either [clockwise (cw)](https://bliptext.com/articles/clockwise) or counterclockwise (ccw) orientation.\nFor example, in mathematics, the reference direction is usually drawn as a [ray](https://bliptext.com/articles/ray-geometry) from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation ([bearing](https://bliptext.com/articles/bearing-navigation), [heading](https://bliptext.com/articles/heading-navigation)) the 0°-[heading](https://bliptext.com/articles/heading-navigation) is drawn vertically upwards and the angle increases for cw rotations.\nThe polar angles decrease towards negative values for rotations in the respectively opposite orientations.\n# [A polar grid with several angles, increasing in counterclockwise orientation and labelled in degrees | upload.wikimedia.org/wikipedia/commons/1/13/Polar///graph///paper.svg]\n\n## Uniqueness of polar coordinates\nAdding any number of full [turns](https://bliptext.com/articles/turn-geometry) (360°) to the angular coordinate does not change the corresponding direction.\nSimilarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle).\nTherefore, the same point (r, φ) can be expressed with an infinite number of different polar coordinates (r, φ + n × 360°) and (−r, φ + 180° + n × 360°) = (−r, φ + (2n + 1) × 180°), where n is an arbitrary [integer](https://bliptext.com/articles/integer).\nMoreover, the pole itself can be expressed as (0, φ) for any angle φ.\nWhere a unique representation is needed for any point besides the pole, it is usual to limit r to positive numbers (r > 0) and φ to either the [interval](https://bliptext.com/articles/interval-mathematics) or the [interval](https://bliptext.com/articles/interval-mathematics), which in radians are or.\nAnother convention, in reference to the usual [codomain](https://bliptext.com/articles/codomain) of the [arctan function](https://bliptext.com/articles/inverse-trigonometric-functions), is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to.\nIn all cases a unique azimuth for the pole (r = 0) must be chosen, e.g., φ = 0.\n\n# Converting between polar and Cartesian coordinates\nThe polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the [trigonometric functions](https://bliptext.com/articles/trigonometric-function) sine and cosine:\nThe Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by:\nwhere hypot is the [Pythagorean sum](https://bliptext.com/articles/pythagorean-addition) and [atan2](https://bliptext.com/articles/atan2) is a common variation on the [arctangent](https://bliptext.com/articles/arctangent) function defined as\nIf r is calculated first as above, then this formula for φ may be stated more simply using the [arccosine](https://bliptext.com/articles/arccosine) function:\n# [A diagram illustrating the relationship between polar and Cartesian coordinates. | upload.wikimedia.org/wikipedia/commons/7/78/Polar///to///cartesian.svg]\n\n## Complex numbers\nEvery [complex number](https://bliptext.com/articles/complex-number) can be represented as a point in the [complex plane](https://bliptext.com/articles/complex-plane), and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form).\nIn polar form, the distance and angle coordinates are often referred to as the number's **magnitude** and **argument** respectively.\nTwo complex numbers can be multiplied by adding their arguments and multiplying their magnitudes.\nThe complex number z can be represented in rectangular form as z = x + iy where i is the [imaginary unit](https://bliptext.com/articles/imaginary-unit), or can alternatively be written in polar form as\nand from there, by [Euler's formula](https://bliptext.com/articles/euler-s-formula), as\nwhere e is [Euler's number](https://bliptext.com/articles/e-mathematical-constant), and φ, expressed in radians, is the [principal value](https://bliptext.com/articles/principal-value) of the complex number function [arg](https://bliptext.com/articles/argument-complex-analysis) applied to x + iy.\nTo convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used.\nEquivalent are the\ncis\nand [angle notations](https://bliptext.com/articles/angle-notation):\nFor the operations of [multiplication](https://bliptext.com/articles/multiplication), [division](https://bliptext.com/articles/division-mathematics), [exponentiation](https://bliptext.com/articles/exponentiation), and [root extraction](https://bliptext.com/articles/root-extraction) of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form.\nFrom the laws of exponentiation:\n# [An illustration of a complex number z plotted on the complex plane | upload.wikimedia.org/wikipedia/commons/7/71/Imaginarynumber2.svg]\n# [An illustration of a complex number plotted on the complex plane using Euler's formula | upload.wikimedia.org/wikipedia/commons/7/71/Euler%27s///formula.svg]\n\n# Polar equation of a curve\nThe equation defining a [plane curve](https://bliptext.com/articles/plane-curve) expressed in polar coordinates is known as a polar equation.\nIn many cases, such an equation can simply be specified by defining r as a [function](https://bliptext.com/articles/function-mathematics) of φ.\nThe resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the [graph](https://bliptext.com/articles/graph-of-a-function) of the polar function r.\nNote that, in contrast to Cartesian coordinates, the independent variable φ is the second entry in the ordered pair.\nDifferent forms of [symmetry](https://bliptext.com/articles/symmetry) can be deduced from the equation of a polar function r:\nr(−φ) = r(φ)\nthe curve will be symmetrical about the horizontal (0°/180°) ray;\nr(π − φ) = r(φ)\nit will be symmetric about the vertical (90°/270°) ray:\nr(φ − α) = r(φ)\nit will be [rotationally symmetric](https://bliptext.com/articles/rotational-symmetry) by α clockwise and counterclockwise about the pole.\nBecause of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate.\nAmong the best known of these curves are the [polar rose](https://bliptext.com/articles/rose-mathematics), [Archimedean spiral](https://bliptext.com/articles/archimedean-spiral), [lemniscate](https://bliptext.com/articles/lemniscate-of-bernoulli), [limaçon](https://bliptext.com/articles/lima-on), and [cardioid](https://bliptext.com/articles/cardioid).\nFor the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.\n\n## Circle\n[[Image:circle r=1.svg|thumb|right|A circle with equation\nr(φ) = 1\n]] The general equation for a circle with a center at and radius a is\nThis can be simplified in various ways, to conform to more specific cases, such as the equation\nfor a circle with a center at the pole and radius a.\nWhen\nr0 = a\nor the origin lies on the circle, the equation becomes\nIn the general case, the equation can be solved for\nr\n, giving\nThe solution with a minus sign in front of the square root gives the same curve.\n\n## Line\nRadial lines (those running through the pole) are represented by the equation\nwhere \\gamma is the angle of elevation of the line; that is,, where m is the [slope](https://bliptext.com/articles/slope) of the line in the Cartesian coordinate system.\nThe non-radial line that crosses the radial line [perpendicularly](https://bliptext.com/articles/perpendicular) at the point has the equation\nOtherwise stated is the point in which the tangent intersects the imaginary circle of radius r_0\n\n## Polar rose\n[[Image:Rose 2sin(4theta).svg|thumb|right|A polar rose with equation\nr(φ) = 2 sin 4φ\n]] A [polar rose](https://bliptext.com/articles/rose-mathematics) is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,\nfor any constant γ0 (including 0).\nIf k is an integer, these equations will produce a k-petaled rose if k is [odd](https://bliptext.com/articles/even-and-odd-numbers), or a 2k-petaled rose if k is even.\nIf k is rational, but not an integer, a rose-like shape may form but with overlapping petals.\nNote that these equations never define a rose with 2, 6, 10, 14, etc. petals.\nThe [variable](https://bliptext.com/articles/variable-math) a directly represents the length or amplitude of the petals of the rose, while k relates to their spatial frequency.\nThe constant γ0 can be regarded as a phase angle.\n\n## Archimedean spiral\n[[File:Spiral of Archimedes.svg|thumb|right|One arm of an Archimedean spiral with equation\nr(φ) = φ / 2π\nfor\n0 0\nand one for\nφ 1, this equation defines a [hyperbola](https://bliptext.com/articles/hyperbola); if\ne = 1\n, it defines a [parabola](https://bliptext.com/articles/parabola); and if\ne < 1\n, it defines an [ellipse](https://bliptext.com/articles/ellipse).\nThe special case\ne = 0\nof the latter results in a circle of the radius \\ell.\n\n## Quadratrix\nA quadratrix in the first quadrant (x, y) is a curve with y = ρ sin θ equal to the fraction of the quarter circle with radius r determined by the radius through the curve point.\nSince this fraction is, the curve is given by.\n\n## Intersection of two polar curves\nThe graphs of two polar functions and have possible intersections of three types:\n\n# Calculus\n[Calculus](https://bliptext.com/articles/calculus) can be applied to equations expressed in polar coordinates.\nThe angular coordinate φ is expressed in radians throughout this section, which is the conventional choice when doing calculus.\n\n## Differential calculus\nUsing\nx = r cos φ\nand\ny = r sin φ\n, one can derive a relationship between derivatives in Cartesian and polar coordinates.\nFor a given function, u(x,y), it follows that (by computing its [total derivatives](https://bliptext.com/articles/total-derivative))\nor\nHence, we have the following formula:\nUsing the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives.\nGiven a function u(r,φ), it follows that\nor\nHence, we have the following formulae:\nTo find the Cartesian slope of the tangent line to a polar curve r(φ) at any given point, the curve is first expressed as a system of [parametric equations](https://bliptext.com/articles/parametric-equations).\n[Differentiating](https://bliptext.com/articles/derivative) both equations with respect to φ yields\nDividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(φ), φ):\nFor other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see [curvilinear coordinates](https://bliptext.com/articles/curvilinear-coordinates).\n\n## Integral calculus (arc length)\nThe arc length (length of a line segment) defined by a polar function is found by the integration over the curve r(φ).\nLet L denote this length along the curve starting from points A through to point B, where these points correspond to φ = a and φ = b such that\n0 < b − a < 2π\n. The length of L is given by the following integral\n\n## Integral calculus (area)\nLet R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π.\nThen, the area of R is\nThis result can be found as follows.\nFirst, the interval [a, b] is divided into n subintervals, where n is some positive integer.\nThus Δφ, the angle measure of each subinterval, is equal to\nb − a\n(the total angle measure of the interval), divided by n, the number of subintervals.\nFor each subinterval i = 1, 2, ..., n, let φi be the midpoint of the subinterval, and construct a [sector](https://bliptext.com/articles/circular-sector) with the center at the pole, radius r(φi), central angle Δφ and arc length r(φi)Δφ.\nThe area of each constructed sector is therefore equal to\nHence, the total area of all of the sectors is\nAs the number of subintervals n is increased, the approximation of the area improves.\nTaking n → ∞, the sum becomes the [Riemann sum](https://bliptext.com/articles/riemann-sum) for the above integral.\nA mechanical device that computes area integrals is the [planimeter](https://bliptext.com/articles/planimeter), which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element [linkage](https://bliptext.com/articles/linkage-mechanical) effects [Green's theorem](https://bliptext.com/articles/green-s-theorem), converting the quadratic polar integral to a linear integral.\n# [polar coordinate system | upload.wikimedia.org/wikipedia/commons/4/4c/Polar///coordinates///integration///region.svg]\n# [The region R is approximated by n sectors (here, n = 5). | upload.wikimedia.org/wikipedia/commons/4/4c/Polar///coordinates///integration///Riemann///sum.svg]\n# [A planimeter, which mechanically computes polar integrals | upload.wikimedia.org/wikipedia/commons/5/54/Planimeter.jpg]\n\n### Generalization\nUsing [Cartesian coordinates](https://bliptext.com/articles/cartesian-coordinates), an infinitesimal area element can be calculated as dA = dx dy.\nThe [substitution rule for multiple integrals](https://bliptext.com/articles/integration-by-substitution) states that, when using other coordinates, the [Jacobian](https://bliptext.com/articles/jacobian-matrix-and-determinant) determinant of the coordinate conversion formula has to be considered:\nHence, an area element in polar coordinates can be written as\nNow, a function, that is given in polar coordinates, can be integrated as follows:\nHere, R is the same region as above, namely, the region enclosed by a curve r(φ) and the rays φ = a and φ = b.\nThe formula for the area of R is retrieved by taking f identically equal to 1.\nA more surprising application of this result yields the [Gaussian integral](https://bliptext.com/articles/gaussian-integral):\n# [A graph of and the area between the function and the x-axis, which is equal to \\sqrt{\\pi}. | upload.wikimedia.org/wikipedia/commons/2/2f/E%5E%28-x%5E2%29.svg]\n\n## Vector calculus\n[Vector calculus](https://bliptext.com/articles/vector-calculus) can also be applied to polar coordinates.\nFor a planar motion, let \\mathbf{r} be the position vector\n(r cos(φ), r sin(φ))\n, with r and\nφ\ndepending on time t.\nWe define an [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) with three unit vectors: radial, transverse, and normal directions.\nThe radial direction is defined by normalizing \\mathbf{r}:\nRadial and velocity directions span the plane of the motion, whose normal direction is denoted :\nThe transverse direction is perpendicular to both radial and normal directions:\nThen\nThis equation can be obtain by taking derivative of the function and derivatives of the unit basis vectors.\nFor a curve in 2D where the parameter is \\theta the previous equations simplify to:\n\n### Centrifugal and Coriolis terms\nThe term is sometimes referred to as the centripetal acceleration, and the term as the Coriolis acceleration.\nFor example, see Shankar.\nNote: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used.\nIn planar particle dynamics these accelerations appear when setting up Newton's [second law of motion](https://bliptext.com/articles/newton-s-second-law) in a rotating frame of reference.\nHere these extra terms are often called [fictitious forces](https://bliptext.com/articles/fictitious-force); fictitious because they are simply a result of a change in coordinate frame.\nThat does not mean they do not exist, rather they exist only in the rotating frame.\n# [Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′ at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ϕ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path. | upload.wikimedia.org/wikipedia/commons/0/0c/Co-rotating///frame///vector.svg]\n\n#### Co-rotating frame\nFor a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous co-rotating frame of reference.\nTo define a co-rotating frame, first an origin is selected from which the distance r(t) to the particle is defined.\nAn axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin.\nThen, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dφ/dt.\nNext, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame.\nLet the location of the particle in the inertial frame be (r(t), φ(t)), and in the co-rotating frame be (r′(t), φ′(t)'').\nBecause the co-rotating frame rotates at the same rate as the particle, dφ′/dt = 0.\nThe fictitious centrifugal force in the co-rotating frame is mrΩ2, radially outward.\nThe velocity of the particle in the co-rotating frame also is radially outward, because dφ′/dt = 0.\nThe fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing φ only.\nThus, using these forces in Newton's second law we find:\nwhere over dots represent derivatives with respect to time, and **F** is the net real force (as opposed to the fictitious forces).\nIn terms of components, this vector equation becomes:\nwhich can be compared to the equations for the inertial frame:\nThis comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dφ/dt, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.\nFor general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous [osculating circle](https://bliptext.com/articles/osculating-circle) of its motion, not to a fixed center of polar coordinates.\nFor more detail, see [centripetal force](https://bliptext.com/articles/centripetal-force).\n\n# Differential geometry\nIn the modern terminology of [differential geometry](https://bliptext.com/articles/differential-geometry), polar coordinates provide [coordinate charts](https://bliptext.com/articles/coordinate-charts) for the [differentiable manifold](https://bliptext.com/articles/differentiable-manifold)\nR2 \\ {(0,0)}\n, the plane minus the origin.\nIn these coordinates, the Euclidean [metric tensor](https://bliptext.com/articles/metric-tensor) is given byThis can be seen via the change of variables formula for the [metric tensor](https://bliptext.com/articles/metric-tensor), or by computing the [differential forms](https://bliptext.com/articles/differential-form) dx, dy via the [exterior derivative](https://bliptext.com/articles/exterior-derivative) of the 0-forms\nx = r cos(θ)\n,\ny = r sin(θ)\nand substituting them in the Euclidean metric tensor\nds2 = dx2 + dy2\n.\nLet, and be two points in the plane given by their cartesian and polar coordinates.\nThen\nSince, and , we get that\nNow we use the trigonometric identity to proceed:\nIf the radial and angular quantities are near to each other, and therefore near to a common quantity r and \\theta, we have that.\nMoreover, the cosine of d\\theta can be approximated with the Taylor series of the cosine up to linear terms:\nso that, and.\nTherefore, around an infinitesimally small domain of any point,\nas stated.\nAn [orthonormal](https://bliptext.com/articles/orthonormality) [frame](https://bliptext.com/articles/moving-frame) with respect to this metric is given bywith dual co[frame](https://bliptext.com/articles/moving-frame)The [connection form](https://bliptext.com/articles/connection-form) relative to this [frame](https://bliptext.com/articles/moving-frame) and the [Levi-Civita connection](https://bliptext.com/articles/levi-civita-connection) is given by the skew-symmetric matrix of 1-formsand hence the [curvature form](https://bliptext.com/articles/curvature-form)\nΩ = dω + ω∧ω\nvanishes.\nTherefore, as expected, the punctured plane is a [flat manifold](https://bliptext.com/articles/flat-manifold).\n\n# Extensions in 3D\nThe polar coordinate system is extended into three dimensions with two different coordinate systems, the [cylindrical](https://bliptext.com/articles/cylindrical-coordinate-system) and [spherical coordinate system](https://bliptext.com/articles/spherical-coordinate-system).\n\n# Applications\nPolar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane.\nThey are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point.\nFor instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate.\nMoreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates.\nThe initial motivation for the introduction of the polar system was the study of [circular](https://bliptext.com/articles/circular-motion) and [orbital motion](https://bliptext.com/articles/orbital-motion).\n\n## Position and navigation\nPolar coordinates are used often in [navigation](https://bliptext.com/articles/navigation) as the destination or direction of travel can be given as an angle and distance from the object being considered.\nFor instance, [aircraft](https://bliptext.com/articles/aircraft) use a slightly modified version of the polar coordinates for navigation.\nIn this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system.\nHeading 360 corresponds to [magnetic north](https://bliptext.com/articles/magnetic-north), while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.\nThus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read [zero-niner-zero](https://bliptext.com/articles/icao-spelling-alphabet) by [air traffic control](https://bliptext.com/articles/air-traffic-control)).\n\n## Modeling\nSystems displaying [radial symmetry](https://bliptext.com/articles/radial-symmetry) provide natural settings for the polar coordinate system, with the central point acting as the pole.\nA prime example of this usage is the [groundwater flow equation](https://bliptext.com/articles/groundwater-flow-equation) when applied to radially symmetric wells.\nSystems with a [radial force](https://bliptext.com/articles/central-force) are also good candidates for the use of the polar coordinate system.\nThese systems include [gravitational fields](https://bliptext.com/articles/gravitation), which obey the [inverse-square law](https://bliptext.com/articles/inverse-square-law), as well as systems with [point sources](https://bliptext.com/articles/point-source), such as [radio antennas](https://bliptext.com/articles/antenna-radio).\nRadially asymmetric systems may also be modeled with polar coordinates.\nFor example, a [microphone](https://bliptext.com/articles/microphone)'s [pickup pattern](https://bliptext.com/articles/microphone-pick-up-patterns) illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves.\nThe curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as at its target design frequency.\nThe pattern shifts toward omnidirectionality at lower frequencies.\n\n## General references","revisionCount":0},"hasMore":false,"nextCursor":null}